Computing Steady Vortex Flows of Prescribed Topology

We consider the problem of finding steady states of the two-dimensional Euler equation from topology-preserving rearrangements of a given vorticity distribution. We begin by briefly reviewing a range of available numerical methodologies. We then focus on a recently introduced technique, which enables the computation of steady vortices with (1) compact vorticity support, (2) prescribed topology, (3) multiple scales, (4) arbitrary stability, and (5) arbitrary symmetry. We highlight a recent set of results, involving the branch of solutions originating at the first bifurcation of the Kirchhoff elliptical vortices. Remarkably, one finds that, as the end of the branch is approached, the family of solutions traces a spiral in a bifurcation diagram involving the rotational velocity and impulse of the configuration. We next show a new set of results, comprising the near-limiting states for a von Kármán street of uniform, finite-area vortices. Building the bifurcation diagram for this flow also reveals a spiral. These and other recent results hint at the possibility that such spirals may constitute a universal feature of families of uniform-vorticity flows.